Optimal. Leaf size=170 \[ \frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tan (c+d x)}{\sqrt {a+b \tan ^2(c+d x)}}\right )}{8 d}+\frac {b \tan (c+d x) \left (a+b \tan ^2(c+d x)\right )^{3/2}}{4 d}+\frac {b (7 a-4 b) \tan (c+d x) \sqrt {a+b \tan ^2(c+d x)}}{8 d}+\frac {(a-b)^{5/2} \tan ^{-1}\left (\frac {\sqrt {a-b} \tan (c+d x)}{\sqrt {a+b \tan ^2(c+d x)}}\right )}{d} \]
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Rubi [A] time = 0.18, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3661, 416, 528, 523, 217, 206, 377, 203} \[ \frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tan (c+d x)}{\sqrt {a+b \tan ^2(c+d x)}}\right )}{8 d}+\frac {(a-b)^{5/2} \tan ^{-1}\left (\frac {\sqrt {a-b} \tan (c+d x)}{\sqrt {a+b \tan ^2(c+d x)}}\right )}{d}+\frac {b \tan (c+d x) \left (a+b \tan ^2(c+d x)\right )^{3/2}}{4 d}+\frac {b (7 a-4 b) \tan (c+d x) \sqrt {a+b \tan ^2(c+d x)}}{8 d} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 217
Rule 377
Rule 416
Rule 523
Rule 528
Rule 3661
Rubi steps
\begin {align*} \int \left (a+b \tan ^2(c+d x)\right )^{5/2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^{5/2}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {b \tan (c+d x) \left (a+b \tan ^2(c+d x)\right )^{3/2}}{4 d}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+b x^2} \left (a (4 a-b)+(7 a-4 b) b x^2\right )}{1+x^2} \, dx,x,\tan (c+d x)\right )}{4 d}\\ &=\frac {(7 a-4 b) b \tan (c+d x) \sqrt {a+b \tan ^2(c+d x)}}{8 d}+\frac {b \tan (c+d x) \left (a+b \tan ^2(c+d x)\right )^{3/2}}{4 d}+\frac {\operatorname {Subst}\left (\int \frac {a \left (8 a^2-9 a b+4 b^2\right )+b \left (15 a^2-20 a b+8 b^2\right ) x^2}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (c+d x)\right )}{8 d}\\ &=\frac {(7 a-4 b) b \tan (c+d x) \sqrt {a+b \tan ^2(c+d x)}}{8 d}+\frac {b \tan (c+d x) \left (a+b \tan ^2(c+d x)\right )^{3/2}}{4 d}+\frac {(a-b)^3 \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (c+d x)\right )}{d}+\frac {\left (b \left (15 a^2-20 a b+8 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\tan (c+d x)\right )}{8 d}\\ &=\frac {(7 a-4 b) b \tan (c+d x) \sqrt {a+b \tan ^2(c+d x)}}{8 d}+\frac {b \tan (c+d x) \left (a+b \tan ^2(c+d x)\right )^{3/2}}{4 d}+\frac {(a-b)^3 \operatorname {Subst}\left (\int \frac {1}{1-(-a+b) x^2} \, dx,x,\frac {\tan (c+d x)}{\sqrt {a+b \tan ^2(c+d x)}}\right )}{d}+\frac {\left (b \left (15 a^2-20 a b+8 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\tan (c+d x)}{\sqrt {a+b \tan ^2(c+d x)}}\right )}{8 d}\\ &=\frac {(a-b)^{5/2} \tan ^{-1}\left (\frac {\sqrt {a-b} \tan (c+d x)}{\sqrt {a+b \tan ^2(c+d x)}}\right )}{d}+\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tan (c+d x)}{\sqrt {a+b \tan ^2(c+d x)}}\right )}{8 d}+\frac {(7 a-4 b) b \tan (c+d x) \sqrt {a+b \tan ^2(c+d x)}}{8 d}+\frac {b \tan (c+d x) \left (a+b \tan ^2(c+d x)\right )^{3/2}}{4 d}\\ \end {align*}
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Mathematica [C] time = 1.34, size = 259, normalized size = 1.52 \[ \frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \log \left (\sqrt {b} \sqrt {a+b \tan ^2(c+d x)}+b \tan (c+d x)\right )+b \tan (c+d x) \sqrt {a+b \tan ^2(c+d x)} \left (9 a+2 b \tan ^2(c+d x)-4 b\right )-4 i (a-b)^{5/2} \log \left (-\frac {4 i \left (\sqrt {a-b} \sqrt {a+b \tan ^2(c+d x)}+a-i b \tan (c+d x)\right )}{(a-b)^{7/2} (\tan (c+d x)+i)}\right )+4 i (a-b)^{5/2} \log \left (\frac {4 i \left (\sqrt {a-b} \sqrt {a+b \tan ^2(c+d x)}+a+i b \tan (c+d x)\right )}{(a-b)^{7/2} (\tan (c+d x)-i)}\right )}{8 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.22, size = 703, normalized size = 4.14 \[ \left [\frac {{\left (15 \, a^{2} - 20 \, a b + 8 \, b^{2}\right )} \sqrt {b} \log \left (2 \, b \tan \left (d x + c\right )^{2} + 2 \, \sqrt {b \tan \left (d x + c\right )^{2} + a} \sqrt {b} \tan \left (d x + c\right ) + a\right ) + 8 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt {-a + b} \log \left (-\frac {{\left (a - 2 \, b\right )} \tan \left (d x + c\right )^{2} + 2 \, \sqrt {b \tan \left (d x + c\right )^{2} + a} \sqrt {-a + b} \tan \left (d x + c\right ) - a}{\tan \left (d x + c\right )^{2} + 1}\right ) + 2 \, {\left (2 \, b^{2} \tan \left (d x + c\right )^{3} + {\left (9 \, a b - 4 \, b^{2}\right )} \tan \left (d x + c\right )\right )} \sqrt {b \tan \left (d x + c\right )^{2} + a}}{16 \, d}, \frac {16 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt {a - b} \arctan \left (-\frac {\sqrt {b \tan \left (d x + c\right )^{2} + a}}{\sqrt {a - b} \tan \left (d x + c\right )}\right ) + {\left (15 \, a^{2} - 20 \, a b + 8 \, b^{2}\right )} \sqrt {b} \log \left (2 \, b \tan \left (d x + c\right )^{2} + 2 \, \sqrt {b \tan \left (d x + c\right )^{2} + a} \sqrt {b} \tan \left (d x + c\right ) + a\right ) + 2 \, {\left (2 \, b^{2} \tan \left (d x + c\right )^{3} + {\left (9 \, a b - 4 \, b^{2}\right )} \tan \left (d x + c\right )\right )} \sqrt {b \tan \left (d x + c\right )^{2} + a}}{16 \, d}, -\frac {{\left (15 \, a^{2} - 20 \, a b + 8 \, b^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b \tan \left (d x + c\right )^{2} + a} \sqrt {-b}}{b \tan \left (d x + c\right )}\right ) - 4 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt {-a + b} \log \left (-\frac {{\left (a - 2 \, b\right )} \tan \left (d x + c\right )^{2} + 2 \, \sqrt {b \tan \left (d x + c\right )^{2} + a} \sqrt {-a + b} \tan \left (d x + c\right ) - a}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (2 \, b^{2} \tan \left (d x + c\right )^{3} + {\left (9 \, a b - 4 \, b^{2}\right )} \tan \left (d x + c\right )\right )} \sqrt {b \tan \left (d x + c\right )^{2} + a}}{8 \, d}, \frac {8 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt {a - b} \arctan \left (-\frac {\sqrt {b \tan \left (d x + c\right )^{2} + a}}{\sqrt {a - b} \tan \left (d x + c\right )}\right ) - {\left (15 \, a^{2} - 20 \, a b + 8 \, b^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b \tan \left (d x + c\right )^{2} + a} \sqrt {-b}}{b \tan \left (d x + c\right )}\right ) + {\left (2 \, b^{2} \tan \left (d x + c\right )^{3} + {\left (9 \, a b - 4 \, b^{2}\right )} \tan \left (d x + c\right )\right )} \sqrt {b \tan \left (d x + c\right )^{2} + a}}{8 \, d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.57, size = 461, normalized size = 2.71 \[ \frac {b^{2} \left (\tan ^{3}\left (d x +c \right )\right ) \sqrt {a +b \left (\tan ^{2}\left (d x +c \right )\right )}}{4 d}+\frac {9 b a \tan \left (d x +c \right ) \sqrt {a +b \left (\tan ^{2}\left (d x +c \right )\right )}}{8 d}+\frac {15 \sqrt {b}\, a^{2} \ln \left (\sqrt {b}\, \tan \left (d x +c \right )+\sqrt {a +b \left (\tan ^{2}\left (d x +c \right )\right )}\right )}{8 d}-\frac {b^{2} \tan \left (d x +c \right ) \sqrt {a +b \left (\tan ^{2}\left (d x +c \right )\right )}}{2 d}-\frac {5 b^{\frac {3}{2}} a \ln \left (\sqrt {b}\, \tan \left (d x +c \right )+\sqrt {a +b \left (\tan ^{2}\left (d x +c \right )\right )}\right )}{2 d}+\frac {b^{\frac {5}{2}} \ln \left (\sqrt {b}\, \tan \left (d x +c \right )+\sqrt {a +b \left (\tan ^{2}\left (d x +c \right )\right )}\right )}{d}-\frac {b \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {\left (a -b \right ) b^{2} \tan \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\tan ^{2}\left (d x +c \right )\right )}}\right )}{d \left (a -b \right )}+\frac {3 a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {\left (a -b \right ) b^{2} \tan \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\tan ^{2}\left (d x +c \right )\right )}}\right )}{d \left (a -b \right )}-\frac {3 a^{2} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {\left (a -b \right ) b^{2} \tan \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\tan ^{2}\left (d x +c \right )\right )}}\right )}{d b \left (a -b \right )}+\frac {a^{3} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {\left (a -b \right ) b^{2} \tan \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\tan ^{2}\left (d x +c \right )\right )}}\right )}{d \,b^{2} \left (a -b \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \tan \left (d x + c\right )^{2} + a\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^2+a\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \tan ^{2}{\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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